Combine the lime juice, onion, jalapeno, pineapple, cilantro, salt and pepper in a small stainless steel, ceramic, glass or metal (you can also just choose one) bowl, tossing occasionally. Let sit at room temperature to marinate, about 15 minutes.
Heat vegetable oil in a medium nonstick skillet over medium-high heat. Add the chorizo and cook, breaking up with a wooden spoon, until browned, about 3 minutes. Drain and set aside.
Preheat a Panini press to medium-high. Brush one side of 2 tortillas with oil and lay dry-side up on a work surface. Scatter each with 1/3 cup cheese. Sandwich with the remaining tortillas and brush the tops with oil.
Place 1 quesadilla in the Panini press at a time and close gently (do not push down). Cook until golden brown and the cheese is melted, 4 to 6 minutes. Repeat with the remaining quesadilla.
Cut the quesadillas into wedges and top with the chorizo and pineapple salsa. Serve with tomato salsa, sour cream and avocado if desired.